What is the definition of $\lim_{h\to 0}f(x,h)=f(x,0)$ uniformly w.r.t x ?
I know uniform convergence of sequences of functions. Does it have anything to do with that?
Is it possible that it is supposed to mean something like the following:
Let $\{h_{n}\}_{n}$ be a sequence s.t. $h_{n}\to 0$, then the sequence of functions defined by $\{f_{n}(x):=f(x,h_{n})\}_{n}$ converges uniformly to $f(x,0)$?
Many thanks in advance!
It means that for any $\varepsilon>0$, there exists $\delta>0$ such that $$ \left|f(x,h)-f(x,0)\right|<\varepsilon $$ whenever $x\in X$ and $0<|h|<\delta$. Note that $\delta$ does not depend on $x$ but depends on $\varepsilon$ only.
For pointwise convergence $\lim_{h\rightarrow0}f(x,h)=f(x,0)$, it means that for each $\varepsilon>0$ and $x\in X$, there exists $\delta>0$ (usually depends on $\varepsilon$ and $x$) such that $\left|f(x,h)-f(x,0)\right|<\varepsilon$ whenever $0<|h|<\delta$.